Rational-driver approximation in car-following theory.
The problem of a car following a lead car driven with constant velocity is considered. To derive the governing equations for the following car dynamics a cost functional is constructed. This functional ranks the outcomes of different driving strategies, which applies to fairly general properties of the driver behavior. Assuming rational-driver behavior, the existence of the Nash equilibrium is proved. Rational driving is defined by supposing that a driver corrects continuously the car motion to follow the optimal path minimizing the cost functional. The corresponding car-following dynamics is described quite generally by a boundary value problem based on the obtained extremal equations. Linearization of these equations around the stationary state results in a generalization of the widely used optimal velocity model. Under certain conditions (the "dense traffic" limit) the rational car dynamics comprises two stages, fast and slow. During the fast stage a driver eliminates the velocity difference between the cars, the subsequent slow stage optimizes the headway. In the dense traffic limit an effective Hamiltonian description is constructed. This allows a more detailed nonlinear analysis. Finally, the differences between rational and bounded rational driver behavior are discussed. The latter, in particular, justifies some basic assumptions used recently by the authors to construct a car-following model lying beyond the frameworks of rationality.